CHAPTER 3 SLUTSKY EQUATION
Economists often are concerned with how a consumer’s behavior changes in response to changes in the economic environment The case we want to consider in this chapter is how a consumer’s choice of a good responds to changes in its price It is natural to think that when the price of a good rises the demand for it will fall However, as we saw in Chapter 6 it is possible to construct examples where the optimal demand for a good decreases when its price falls A good that has this property is called a Giffen good
Giffen goods are pretty peculiar and are primarily a theoretical curiosity, but there are other situations where changes in prices might have “perverse” effects that, on reflection, turn out not to be so unreasonable For example, we normally think that if people get a higher wage they will work more But what if your wage went from $10 an hour to $1000 an hour? Would you really work more? Might you not decide to work fewer hours and use some of the money you’ve earned to do other things? What if your wage were $1,000,000 an hour? Wouldn’t you work less?
Trang 2THE SUBSTITUTION EFFECT 137
how about a family who grew apples to sell? If the price of apples went up, their income might go up so much that they would feel that they could now afford to consume more of their own apples For the consumers in this family, an increase in the price of apples might well lead to an increase in the consumption of apples
What is going on here? How is it that changes in price can have these ambiguous effects on demand? In this chapter and the next we'll try to sort out these effects
8.1 The Substitution Effect
When the price of a good changes, there are two sorts of effects: the rate at which you can exchange one good for another changes, and the total purchasing power of your income is altered If, for example, good 1 becomes cheaper, it means that you have to give up less of good 2 to purchase good 1 The change in the price of good 1 has changed the rate at which the market allows you to “substitute” good 2 for good 1 The trade-off between the two goods that the market presents the consumer has changed
At the same time, if good 1 becomes cheaper it means that your money income will buy more of good 1 The purchasing power of your money has gone up; although the number of dollars you have is the same, the amount that they will buy has increased
The first part—the change in demand due to the change in the rate of exchange between the two goods—is called the substitution effect The second effect—the change in demand due to having more purchasing power—is called the income effect These are only rough definitions of the two effects In order to give a more precise definition we have to consider the two effects in greater detail
The way that we will do this is to break the price movement into two steps: first we will let the relatzve prices change and adjust money income so as to hold purchasing power constant, then we will let purchasing power adjust while holding the relative prices constant
This is best explained by referring to Figure 8.1 Here we have a situa- tion where the price of good 1 has declined This means that the budget line rotates around the vertical intercept m/p2 and becomes flatter We can break this movement of the budget line up into two steps: first pivot the budget line around the original demanded bundle and then shift the pivoted line out to the new demanded bundle
Trang 3138 SLUTSKY EQUATION (Ch 8) xy Indifference curves Original budget Original choice Final choice Pivoted budget Shift XI x Final
Pivot and shift When the price of good 1: changes and income stays fixed, the budget line pivots around the vertical axis We
will view this adjustment as occurring in two stages: first pivot
the budget line around the original ‘choice,-and then shift this line outward to the new demanded ‘bundle
in price and chooses a new bundle of goods in response But in analyzing how the consumer’s choice changes, it is useful to think of the budget line changing in two stages—first the pivot, then the shift
What are the economic meanings of the pivoted and the shifted budget lines? Let us first consider the pivoted line Here we have a budget line with the same slope and thus the same relative prices as the final budget line However, the money income associated with this budget line is different,
since the vertical intercept is different Since the original consumption
bundle (21,22) lies on the pivoted budget line, that consumption bundle is just affordable The purchasing power of the consumer has remained constant in the sense that the original bundle of goods is just affordable at the new pivoted line
Let us calculate how much we have to adjust money income in order to keep the old bundle just affordable Let m’ be the amount of money income that will just make the original consumption bundle affordable; this will be the amount of money income associated with the pivoted budget line Since (71, £2) is affordable at both (pi, po,m) and (p{,pe,m’), we have
m’ = pix, + pee
Mm = Py + peta
Subtracting the second equation from the first gives
Trang 4THE SUBSTITUTION EFFECT 139
This equation says that the change in money income necessary to make the old bundle affordable at the new prices is just the original amount of consumption of good 1 times the change in prices
Letting Ap; = pi — pi represent the change in price 1, and Am = m —m represent the change in income necessary to make the old bundle just affordable, we have
Am = 2, Apr (8.1)
Note that the change in income and the change in price will always move in the same direction: if the price goes up, then we have to raise income to keep the same bundle affordable
Let’s use some actual numbers Suppose that the consumer is originally consuming 20 candy bars a week, and that candy bars cost 50 cents a piece If the price of candy bars goes up by 10 cents—so that Ap; = 60 — 50 = 10—how much would income have to change to make the old consumption bundle affordable?
We can apply the formula given above If the consumer had $2.00 more income, he would just be able to consume the same number of candy bars, namely, 20 In terms of the formula:
Am = Ap, X= 10 x 20 = $2.00
Now we have a formula for the pivoted budget line: it is just the budget line at the new price with income changed by Am Note that if the price of good 1 goes down, then the adjustment in income will be negative When a price goes down, a consumer’s purchasing power goes up, so we will have to decrease the consumer’s income in order to keep purchasing power fixed Similarly, when a price goes up, purchasing power goes down, so the change in income necessary to keep purchasing power constant must be positive
Although (21, 22) is still affordable, it is not generally the optimal pur-
chase at the pivoted budget line In Figure 8.2 we have denoted the optimal
purchase on the pivoted budget line by Y This bundle of goods is the op-
timal bundle of goods when we change the price and then adjust dollar income so as to keep the old bundle of goods just affordable The move- ment from X to Y is known as the substitution effect It indicates how the consumer “substitutes” one good for the other when a price changes but purchasing power remains constant
More precisely, the substitution effect, Ag}, is the change in the demand for good 1 when the price of good 1 changes to p; and, at the same time, money income changes to m’:
Azi = #i(p,m) — #1(pì, mm)
In order to determine the substitution effect, we must use the consumer’s
demand function to calculate the optimal choices at (p,,m’) and (p, m)
Trang 5140 SLUTSKY EQUATION (Ch 8) Indifference curves m'/p, Pivot —————— x Substitution Income effect effect
Substitution effect and income effect The pivot gives the substitution effect, and the shift gives the income effect
on the shape of the consumer’s indifference curves But given the demand function, it is easy to just plug in the numbers to calculate the substitution
effect (Of course the demand for good 1 may well depend on the price of
good 2; but the price of good 2 is being held constant during this exercise, so we’ve left it out of the demand function so as not to clutter the notation.) The substitution effect is sometimes called the change in compensated demand The idea is that the consumer is being compensated for a price rise by having enough income given back to him to purchase his old bun- dle Of course if the price goes down he is “compensated” by having money taken away from him We’ll generally stick with the “substitution” termi- nology, for consistency, but the “compensation” terminology is also widely used
EXAMPLE: Calculating the Substitution Effect
Suppose that the consumer has a demand function for milk of the form
Trang 6
THE INCOME EFFECT 141
Now suppose that the price of milk falls to $2 per quart Then his
demand at this new price will be 10 + 120/(10 x 2) = 16 quarts of milk per
week, The total change in demand is +2 quarts a week
In order to calculate the substitution effect, we must first calculate how
much income would have to change in order to make the original consump- tion of milk just affordable when the price of milk is $2 a quart We apply
the formula (8.1):
Am = x Ap, = l4x (2 — 3) = —814
Thus the level of income necessary to keep purchasing power constant ism’ = m+ Am = 120-14 = 106 What is the consumer’s demand for milk at the new price, $2 per quart, and this level of income? Just plug the numbers into the demand function to find 106 10 x 2 ai (pj, m’) = 21 (2, 106) = 10 + = 15.3 Thus the substitution effect is Ax? = x1(2,106) — z¡(3,120) = 15.3 — 14 = 1.3
8.2 The Income Effect
We turn now to the second stage of the price adjustment—the shift move- ment This is also easy to interpret economically We know that a parallel shift of the budget line is the movement that occurs when income changes while relative prices remain constant Thus the second stage of the price adjustment is called the income effect We simply change the consumer’s income from m’ to m, keeping the prices constant at (p,p2) In Figure
8.2 this change moves us from the point (y1, y2) to (21, 22) Ít is natural to
call this last movement the income effect since all we are doing is changing income while keeping the prices fixed at the new prices
More precisely, the income effect, Az’, is the change in the demand for good 1 when we change income from m’ to m, holding the price of good 1
fixed at pj:
Ag? = a1(p,,m) — 21(p4,m’)
We have already considered the income effect earlier in section 6.1 There we saw that the income effect can operate either way: it will tend to increase or decrease the demand for good 1 depending on whether we have a normal good or an inferior good
When the price of a good decreases, we need to decrease income in order
Trang 7142 SLUTSKY EQUATION (Ch, 8)
EXAMPLE: Calculating the Income Effect
In the example given earlier in this chapter we saw that
#1(01,m) = #¡(2, 120) = 16
x1(pi,m’) = x1(2,106) = 15.3
Thus the income effect for this problem is
Ax? = 1(2, 120) — a, (2,106) = 16 ~ 15.3 = 0.7
Since milk is a normal good for this consumer, the demand for milk in- creases when income increases
8.3 Sign of the Substitution Effect
We have seen above that the income effect can be positive or negative, de- pending on whether the good is a normal good or an inferior good What about the substitution effect? If the price of a good goes down, as in Figure 8.2, then the change in the demand for the good due to the substi- tution effect must be nonnegative That is, if p; > pi, then we must have
#1(p1,rn') > x1(pi,m), so that Azj > 0
The proof of this goes as follows Consider the points on the pivoted budget line in Figure 8.2 where the amount of good 1 consumed is less than at the bundle X These bundles were all affordable at the old prices (p1,p2} but they weren’t purchased Instead the bundle X was purchased If the consumer is always choosing the best bundle he can afford, then X must be preferred to all of the bundles on the part of the pivoted line that lies inside the original budget set
This means that the optimal choice on the pivoted budget line must not
be one of the bundles that lies underneath the original budget line The
optimal choice on the pivoted line would have to be either X or some point to the right of X: But this means that the new optimal choice must involve consuming at least as much of good 1 as originally, just as we wanted to show In the case illustrated in Figure 8.2, the optimal choice at the pivoted budget line is the bundle Y, which certainly involves consuming more of good 1 than at the original consumption point, X
Trang 8THE TOTAL CHANGE IN DEMAND — 143
8.4 The Total Change in Demand
The total change in demand, Az, is the change in demand due to the change in price, holding income constant:
AZ = #1(01,m) — 21(pi,m)
We have seen above how this change can be broken up into two changes: the substitution effect and the income effect In terms of the symbols defined
above,
Ag, = Ary + Az}
21(p,,m) — £1(p1,m) = [z1(p,,m’) ~ x1(pi, m)] + [xi(p,m) ~ 21(pi,m’)]
In words this equation says that the total change in demand equals the substitution effect plus the income effect This equation is called the Slut- sky identity.! Note that it is an identity: it is true for all values of py, py, m, and m’ The first and fourth terms on the right-hand side cancel out, so the right-hand side is identically equal to the left-hand side
The content of the Slutsky identity is not just the algebraic identity— that is a mathematical triviality The content comes in the interpretation of the two terms on the right-hand side: the substitution effect and the income effect In particular, we can use what we know about the signs of the income and substitution effects to determine the sign of the total effect
Trang 9144 SLUTSKY EQUATION (Ch 8)
Note carefully the sign on the income effect Since we are considering a situation where the price rises, this implies a decrease in purchasing power—for a normal good this will imply a decrease in demand
On the other hand, if we have an inferior good, it might happen that the income effect outweighs the substitution effect, so that the total change in demand associated with a price increase is actually positive This would
be a case where
Am = Azj + Az† @ () G)
If the second term on the right-hand side—the income effect—is large enough, the total change in demand could be positive This would mean that an increase in price could result in an increase in demand This is the perverse Giffen case described earlier: the increase in price has reduced the consumer’s purchasing power so much that he has increased his consump- tion of the inferior good
But the Slutsky identity shows that this kind of perverse effect can only occur for inferior goods: if a good is a normal good, then the income and substitution effects reinforce each other, so that the total change in demand is always in the “right” direction
Thus a Giffen good must be an inferior good But an inferior good is not necessarily a Giffen good: the income effect not only has to be of the “wrong” sign, it also has to be large enough to outweigh the “right” sign of the substitution effect This is why Giffen goods are so rarely observed in real life: they would not only have to be inferior goods, but they would have to be very inferior
This is illustrated graphically in Figure 8.3 Here we illustrate the usual pivot-shift operation to find the substitution effect and the income effect In both cases, good 1 is an inferior good, and the income effect is therefore negative In Figure 8.3A, the income effect is large enough to outweigh the substitution effect and produce a Giffen good In Figure 8.3B, the income effect is smaller, and thus good 1 responds in the ordinary way to the change in its price
8.5 Rates of Change
We have seen that the income and substitution effects can be described
graphically as a combination of pivots and shifts, or they can be described algebraically in the Slutsky identity
Azi = Azj + AzT†,
Trang 10RATES OF CHANGE 145 x2 Indifference curves Original budget line curves indifference Original budget line budget Lgl | Li } i L sags xt os xy Income Substitution income — Substitution | Total Total
A The Giffen case B Non-Giffen inferior good
Inferior goods Panel A shows a good that is inferior enough to cause the Giffen case Panel B shows a good that is inferior, but the effect is not strong enough to create a Giffen good
8.3
of absolute changes, but it is more common to express it in terms of rates of change
When we express the Slutsky identity in terms of rates of change it turns
out to be convenient to define Ax?” to be the negative of the income effect:
Art = #1(m,m') — #1(1,rn) = ~—Azy
Given this definition, the Slutsky identity becomes Ag, = Az} — Aat’
If we divide each side of the identity by Api, we have
Azi
= 8.2
Api Api ( )
The first term on the right-hand side is the rate of change of demand when price changes and income is adjusted so as to keep the old bundle affordable—the substitution effect Let’s work on the second term Since we have an income change in the numerator, it would be nice to get an income change in the denominator
bate
Trang 11146 SLUTSKY EQUATION (Ch 8)
Remember that the income change, Am, and the price change, Ap), are related by the formula Am = x, Ap Solving for Ap, we find Am Ap, = — +1 Now substitute this expression into the last term in (8.2) to get our final formula: Ax, _ Axvi Az? Ap, Ap, Am mm
This is the Slutsky identity in terms of rates of change We can interpret each term as follows:
Ar, _ t1(pi,m) ~ 21(pi,m)
Ap Ap,
is the rate of change in demand as price changes, holding income fixed;
Azj _ ziÚ1,1n)) ~ ziÚn,1n)
Am Am
is the rate of change in demand as the price changes, adjusting income so as to keep the old bundle just affordable, that is, the substitution effect; and
Ac? _ miứ,m')— zi(p,m) T1 = #1
Am m — rn
(8.3)
is the rate of change of demand holding prices fixed and adjusting income,
that is, the income effect
The income effect is itself composed of two pieces: how demand changes as income changes, times the original level of demand When the price
changes by Ap,, the change in demand due to the income effect is x1 (pi,m’) — #1 (p1,m)
Agy’ = Am xi Apr
But this last term, x, Apj, is just the change in income necessary to keep the old bundle feasible That is, rz; Ap; = Am, so the change in demand due to the income effect reduces to
, \ ,
Ag® = #1(p1,Tn "<= Am,
Trang 12EXAMPLES OF INCOME AND SUBSTITUTION EFFECTS 147
8.6 The Law of Demand
In Chapter 5 we voiced some concerns over the fact that consumer theory seemed to have no particular content: demand could go up or down when a price increased, and demand could go up or down when income increased If a theory doesn’t restrict observed behavior in some fashion it isn’t much of atheory A model that is consistent with all behavior has no real content However, we know that consumer theory does have some content—we’ve seen that choices generated by an optimizing consumer must satisfy the
Strong Axiom of Revealed Preference Furthermore, we’ve seen that any
price change can be decomposed into two changes: a substitution effect that is sure to be negative—opposite the direction of the price change— and an income effect whose sign depends on whether the good is a normal good or an inferior good
Although consumer theory doesn’t restrict how demand changes when
price changes or how demand changes when income changes, it does re- strict how these two kinds of changes interact In particular, we have the following
The Law of Demand If the demand for a good increases when income increases, then the demand for that good must decrease when its price in-
creases
This follows directly from the Slutsky equation: if the demand increases when income increases, we have a normal good And if we have a normal
good, then the substitution effect and the income effect reinforce each other,
and an increase in price will unambiguously reduce demand
8.7 Examples of Income and Substitution Effects
Let’s now consider some examples of price changes for particular kinds of preferences and decompose the demand changes into the income and the substitution effects
We start with the case of perfect complements The Slutsky decomposi- tion is illustrated in Figure 8.4 When we pivot the budget line around the chosen point, the optimal choice at the new budget line is the same as at the old one—this means that the substitution effect is zero The change in demand is due entirely to the income effect
Trang 13148 SLUTSKY EQUATION (Ch 8) xy indifference curves Original budget line Final budget line -— - F —
Income effect = total effect
Perfect complements Slutsky decomposition with perfect complements
As a third example, let us consider the case of quasilinear preferences This situation is somewhat peculiar We have already seen that a shift in income causes no change in demand for good 1 when preferences are quasilinear This means that the entire change in demand for good 1 is due to the substitution effect, and that the income effect is zero, as illustrated in Figure 8.6
EXAMPLE: Rebating a Tax
In 1974 the Organization of Petroleum Exporting Countries (OPEC) insti- tuted an oil embargo against the United States OPEC was able to stop oil shipments to U.S ports for several weeks The vulnerability of the United States to such disruptions was very disturbing to Congress and the pres- ident, and there were many plans proposed to reduce the United States’s dependence on foreign oil
One such plan involved increasing the gasoline tax Increasing the cost of gasoline to the consumers would make them reduce their consumption of gasoline, and the reduced demand for gasoline would in turn reduce the demand for foreign oil
Trang 14EXAMPLES OF INCOME AND SUBSTITUTION EFFECTS 149 indifference curves Original
choice \ Final budget line
Original Final choice
budget
line |
Substitution effect = total effect
Perfect substitutes Slutsky decomposition with perfect sub- stitutes
infeasible So it was suggested that the revenues raised from consumers by this tax would be returned to the consumers in the form of direct money payments, or via the reduction of some other tax
Critics of this proposal argued that paying the revenue raised by the tax back to the consumers would have no effect on demand since they could just use the rebated money to purchase more gasoline What does economic analysis say about this plan?
Let us suppose, for simplicity, that the tax on gasoline would end up being passed along entirely to the consumers of gasoline so that the price of gasoline will go up by exactly the amount of the tax (In general, only part of the tax would be passed along, but we will ignore that complication here.) Suppose that the tax would raise the price of gasoline from p to p’ = p+t, and that the average consumer would respond by reducing his demand from z to x’ The average consumer is paying ¢ dollars more for gasoline, and he is consuming 2’ gallons of gasoline after the tax is imposed, so the amount of revenue raised by the tax from the average
consumer would be
R=ta' = (p'—p)a’
Trang 15150 SLUTSKY EQUATION (Ch 8) Indifference Original budget line Pivot
Substitution effect = total effect
Quasilinear preferences In the case of quasilinear prefer-
ences, the entire change in demand is due to the substitution effect
consuming, ©
If we let y be the expenditure on all other goods and set its price to be 1, then the original budget constraint is
pe +y =m, (8.4)
and the budget constraint in the presence of the tax-rebate plan is
(p+t†)z' + =m + ta (8.5)
In budget constraint (8.5) the average consumer is choosing the left-hand side variables—the consumption of each good—but the right-hand side— his income and the rebate from the government—are taken as fixed The rebate depends on what all consumers do, not what the average consumer does In this case, the rebate turns out to be the taxes collected from the average consumer—but that’s because he is average, not because of any causal connection
If we cancel tz’ from each side of equation (8.5), we have
pa' + ` = m
Trang 16EXAMPLES OF INCOME AND SUBSTITUTION EFFECTS 151
is preferred to (#’,y’): the consumers are made worse off by this plan
Perhaps that is why it was never put into effect!
The equilibrium with a rebated tax is depicted in Figure 8.7 The tax makes good 1 more expensive, and the rebate increases money income The original bundle is no longer affordable, and the consumer is definitely made worse off The consumer’s choice under the tax-rebate plan involves consuming less gasoline and more of “all other goods.” Indifference curves xt’) Budget line after tax and rebate slope =—(p + t) Budget line before tax slope =— p
Rebating a tax Taxing a consumer and rebating the tax
revenues makes the consumer worse off
What can we say about the amount of consumption of gasoline? The average consumer could afford his old consumption of gasoline, but because of the tax, gasoline is now more expensive In general, the consumer would choose to consume less of it
EXAMPLE: Voluntary Real Time Pricing
Trang 17152 SLUTSKY EQUATION (Ch 8)
finding ways to reduce the use of electricity during periods of peak demand is very attractive from an economic point of view
In states with warm climates, such as Georgia, roughly 30 percent of usage during periods of peak demand is due to air conditioning Further- more, it is relatively easy to forecast temperature one day ahead so that potential users will have time to adjust their demand by setting their air conditioning to a higher temperature, wearing light clothes, and so on, The challenge is to set up a pricing system so that those users who are able to cut back on their electricity use will have an incentive to reduce their consumption
One way to accomplish this is through the use of Real Time Pricing (RTP) In a Real Time Pricing program, large industrial users are equipped with special meters that, allow the price of electricity to vary from minute to minute, depending on signals sent from the electricity generating company As the demand for electricity approaches capacity, the generating company increases the price so as to encourage users to cut back on their usage The price schedule is determined as a function of the total demand for electricity
Georgia Power Company claims that it runs the largest real time pric- ing program in the world In 1999 it was able to reduce demand by 750 megawatts on high-price days by inducing some large customers to cut their demand by as much as 60 percent
Georgia Power has devised several interesting variations on the basic real time pricing model In one pricing plan, customers are assigned a baseline quantity, which represents their normal usage When electricity is in short supply and the real time price increases, these users face a higher price for electricity use in excess of their baseline quantity But they also receive a rebate if they can manage to cut their electricity use below their baseline
amount
Figure 8.8 shows how this affects the budget line of the users The
vertical axis is “money to spend on things other than electricity” and the horizontal axis is “electricity use.” In normal times, users choose their electricity consumption to maximize utility subject to a budget constraint which is determined by the baseline price of electricity The resulting choice is their baseline consumption
When the temperature rises, the real time price increases, making elec- tricity more expensive But this increase in price is a good thing for users who can cut back their consumption, since they receive a rebate based on the high real time price for every kilowatt of reduced usage If usage stays at the baseline amount, then the user’s bill will not change
Trang 18ANOTHER SUBSTITUTION EFFECT 153 OTHER GOODS Consumption under-RTP RTP budget constraint Baseline consumption _— Baseline budget constraint ELECTRICITY
Voluntary real time pricing Users pay higher rates for
additional electricity when the real time price rises, but they
also get rebates at the same price if they cut back their use
This results in a pivot around the baseline use and tends to make the customers better off
8.8 Another Substitution Effect
The substitution effect is the name that economists give to the change in demand when prices change but a consumer’s purchasing power is held constant, so that the original bundle remains affordable At least this is one definition of the substitution effect There is another definition that is also useful
The definition we have studied above is called the Slutsky substitution effect The definition we will describe in this section is called the Hicks
substitution effect.”
Suppose that instead of pivoting the budget line around the original consumption bundle, we now roll the budget line around the indifference curve through the original consumption bundle, as depicted in Figure 8.9 In this way we present the consumer with a new budget line that has the same relative prices as the final budget line but has a different income The purchasing power he has under this budget line will no longer be sufficient to
Trang 19154 SLUTSKY EQUATION (Ch 8)
purchase his original bundle of goods—but it will be sufficient to purchase a bundle that is just indifferent to his original bundle Indifference curves Original budget Original choice Substitution - Income effect effect
The Hicks substitution effect Here we pivot the budget line around the indifference curve rather than around the original
choice
Thus the Hicks substitution effect keeps utility constant rather than keep- ing purchasing power constant The Slutsky substitution effect gives the consumer just enough money to get back to his old level of consumption, while the Hicks substitution effect gives the consumer just enough money to get back to his old indifference curve Despite this difference in defini- tion, it turns out that the Hicks substitution effect must be negative—in
the sense that it is in a direction opposite that of the price change—just
like the Slutsky substitution effect
The proof is again by revealed preference Let (11,22) be a demanded bundle at some prices (p;,p2), and let (yi, y2) be a demanded bundle at
some other prices (q1,q2) Suppose that income is such that the consumer is indifferent between (21,72) and (y;, y2) Since the consumer is indifferent between (21,22) and (yi, ye), neither bundle can be revealed preferred to the other
Trang 20COMPENSATED DEMAND CURVES — 155
two inequalities are not true:
Pity + poke > pry + p2ye 911 + đ2U2 > điZ1 + q24 It follows that these inequalities are true:
Pit, + Pete S piys + poye
gayi + đ292 Š điZ1 + 4242
Adding these inequalities together and rearranging them we have
(gì — Đi)(Uì — #1) + (ga — pa) (ye — 22) < 0
This is a general statement about how demands change when prices change if income is adjusted so as to keep the consumer on the same in- difference curve In the particular case we are concerned with, we are only changing the first price Therefore gz = po, and we are left with
(gi — p1)(/¡ — #1) <0
This equation says that the change in the quantity demanded must have the opposite sign from that of the price change, which is what we wanted to show
The total change in demand is still equal to the substitution effect plus the income effect—but now it is the Hicks substitution effect Since the Hicks substitution effect is also negative, the Slutsky equation takes exactly the same form as we had earlier and has exactly the same interpretation Both the Slutsky and Hicks definitions of the substitution effect have their place, and which is more useful depends on the problem at hand It can be shown that for small changes in price, the two substitution effects are virtually identical
8.9 Compensated Demand Curves
We have seen how the quantity demanded changes as a price changes in
three different contexts: holding income fixed (the standard case), holding
purchasing power fixed (the Slutsky substitution effect), and holding utility
fixed (the Hicks substitution effect) We can draw the relationship between
price and quantity demanded holding any of these three variables fixed This gives rise to three different demand curves: the standard demand curve, the Slutsky demand curve, and the Hicks demand curve
Trang 21156 SLUTSKY EQUATION (Ch 8)
demand curve is a downward sloping curve for normal goods However, the Giffen analysis shows that it is theoretically possible that the ordinary demand curve may slope upwards for an inferior good
The Hicksian demand curve—the one with utility held constant—is some- times called the compensated demand curve This terminology arises naturally if you think of constructing the Hicksian demand curve by ad- justing income as the price changes so as to keep the consumer’s utility constant Hence the consumer is “compensated” for the price changes, and his utility is the same at every point on the Hicksian demand curve This is in contrast to the situation with an ordinary demand curve In this case the consumer is worse off facing higher prices than lower prices since his income is constant
The compensated demand curve turns out to be very useful in advanced courses, especially in treatments of benefit-cost analysis In this sort of analysis it is natural to ask what size payments are necessary to compen- sate consumers for some policy change The magnitude of such payments gives a useful estimate of the cost of the policy change However, actual calculation of compensated demand curves requires more mathematical ma- chinery than we have developed in this text
Summary
1 When the price of a good decreases, there will be two effects on consump- tion The change in relative prices makes the consumer want to consume more of the cheaper good The increase in purchasing power due to the
lower price may increase or decrease consumption, depending on whether
the good is a normal good or an inferior good
2 The change in demand due to the change in relative prices is called the substitution effect; the change due to the change in purchasing power is called the income effect
3 The substitution effect is how demand changes when prices change and purchasing power is held constant, in the sense that the original bundle remains affordable To hold real purchasing power constant, money income will have to change The necessary change in money income is given by
Am = 2, Ap}
4, The Slutsky equation says that the total change in demand is the sum
of the substitution effect and the income effect
Trang 22APPENDIX 157
REVIEW QUESTIONS
1 Suppose a consumer has preferences between two goods that are perfect
substitutes Can you change prices in such a way that the entire demand response is due to the income effect?
2 Suppose that preferences are concave Is it still the case that the substi- tution effect is negative?
3 In the case of the gasoline tax, what would happen if the rebate to the consumers were based on their original consumption of gasoline, x, rather than on their final consumption of gasoline, 2’?
4 In the case described in the preceding question, would the government be paying out more or less than it received in tax revenues?
5 In this case would the consumers be better off or worse off if the tax with rebate based on original consumption were in effect?
APPENDIX
Let us derive the Slutsky equation using calculus Consider the Slutsky defini- tion of the substitution effect, in which the income is adjusted so as to give the consumer just enough to buy the original consumption bundle, which we will now denote by (%1,%2) If the prices are (pi, p2), then the consumer’s actual choice
with this adjustment will depend on (pi, p2) and (#1, £2) Let’s call this relation-
ship the Stutsky demand function for good 1, and write it as xj (p1, pe, £1, T2) Suppose the original demanded bundle is (£1, 2) at prices (p,,P.) and income m The Slutsky demand function tells us what the consumer would demand facing some different prices (pi,p2) and having income p1%1 + peX2 Thus the
Slutsky demand function at (p1, p2,%1, ¥2) is the ordinary demand at (p1,p2) and
income pizx1 + pe¥2 That is,
24 (pr, P2, £1, £2) = t1(p1, p2, pr Fi + p22)
This equation says that the Slutsky demand at prices (p1,p2) is that amount which the consumer would demand if he had enough income to purchase his
original bundle of goods (%1, £2) This is just the definition of the Slutsky demand
function
Differentiating this identity with respect to pi, we have
Ox} (Piy P2,F1,F2) _ O21 (pi, p2, m7) + G1 (pi, p2, 77)
Opi Op, ôm,
Rearranging we have
Oxi (Pi, pa, 7) _ Oxi (p1, p2, £1, €2) _ 981(P1, Pas)
Trang 23158 SLUTSKY EQUATION (Ch 8)
Note the use of the chain rule in this calculation
This is a derivative form of the Slutsky equation It says that the total effect of a price change is composed of a substitution effect (where income is adjusted to keep the bundle (%1,%2) feasible) and an income effect We know from the text that the substitution effect is negative and that the sign of the income effect depends on whether the good in question is inferior or not As you can see, this is just the form of the Slutsky equation considered in the text, except that we have replaced the A’s with derivative signs
What about the Hicks substitution effect? It is also possible to define a Slutsky equation for it We let tt (pi, po, %) be the Hicksian demand function, which measures how much the consumer demands of good 1 at prices (pi, p2) if income is adjusted to keep the level of utility constant at the original level % It turns out that in this case the Slutsky equation takes the form
Oxi(pi,p2,m) _ Out (pryp2,%) _ Ox1(pr,p2,m) „
Opi Op, om tụ
The proof of this equation hinges on the fact that
dx} (pi, p2,%) _ Ox} (pi, p2,F1, £2)
Opi Opi
for infinitesimal changes in price That is, for derivative size changes in price, the Slutsky substitution and the Hicks substitution effect are the same The proof of this is not terribly difficult, but it involves some concepts that are beyond the scope of this book A relatively simple proof is given in Hal R Varian, Microeconomic Analysis, 3rd ed (New York: Norton, 1992)
EXAMPLE: Rebating a Small Tax
We can use the calculus version of the Slutsky equation to see how consumption choices would react to a small change in a tax when the tax revenues are rebated to the consumers
Assume, as before, that the tax causes the price to rise by the full amount of the tax Let x be the amount of gasoline, p its original price, and t the amount of the tax Then the change in consumption will be given by
The first term measures how demand responds to the price change times the amount of the price change—which gives us the price effect of the tax The second terms tells us how demand responds to a change in income times the amount that income has changed—income has gone up by the amount of the tax revenues rebated to the consumer
Now use Slutsky’s equation to expand the first term on the right-hand side to get the substitution and income effects of the price change itself:
Trang 24APPENDIX 159 The income effect cancels out, and all that is left is the pure substitution effect